Minimal free resolutions for subschemes of star configurations
Let \(R=K[x_1,\dots,x_n]\). Let \(C_{c,n}\) denote the set of subsets of \(\{1,2,\dots,n\}\) with cardinality \(c\). For a subset \(S\) of \(C_{c,n}\), one can associate a squarefree monomial ideal \(I_S=\bigcap_{\{a_1,\dots,a_c\}\in S} (x_{a_1},\dots,x_{a_c})\) in \(R\) and vice versa. In particular, \(I_{C_{c,n}}\) defines the \textit{star configuration}. In the paper under review, the authors investigate the subschemes of a star configuration whose support \(S_{n,h}\) is a union of \(n-2h\) consecutive diagonals \(D_k\), where \(D_k=\{\{i,j\}\in C_{2,n} \mid |i-j|=k\}\) and \(S_{n,h}\) is defined recursively as follows: \(S_{n,h}=S_{n,h-1}\setminus (D_h\cup D_{n-h}), 2 \leq h \leq \frac{n-2}{2}\) with \(S_{n,1}=C_{2,n}\setminus P_n\), where \(P_n=\{ \{i,j\}\in C_{2,n}\mid |i-j|=1 \} \cup \{\{1,n\}\}\). The authors compute the minimal graded free resolutions of \(I_{S_{n,h}}\) for various values of \(h\) (Corollary 2.11, Theorem 3.6).
scientific article; zbMATH DE number 6499146
scientific article